T-forward measure.

he measure associated with the
numeraire
is called the
"
forward
measure". It is particularly useful when evaluating a price of a derivative.
Indeed, the regular risk neutral measure corresponds to the defined above, see
(
MMA numeraire
), numeraire
,
and
Hence, the transformation to the T-forward measure moves the discounting
outside of the expectation term.

Suppose an asset
and the riskless bond
are given by the
equations
with respect to the risk neutral measure. Here
and
are correlated increments of the standard Brownian
motions,
We perform transformation of the SDEs to the T-forward measure. The old
numeraire is
:
and the new numeraire is
.
Hence, according to the formula (
Change of
drift recipe
), the drift of
in the
-measure
is
Therefore
where the
is increment of the standard Brownian motion with respect to the
-forward
probability.